Why does background image disappear, UNCHECKED - when the 3d view is closed, re-opened, or duplicated?
The background image settings are saved to the 3DView that the properties region is attached to. By changing another editor to a 3DView a new set of default values are used, they don't get copied from another 3DView.

If you setup the background images and then split the 3DView, then the background settings will get copied to the new editor.